The University of Arizona
Please note that this event has ended!

Factorisation Dynamics and Total Positivity at Several Scales

Analysis, Dynamics, and Applications Seminar

Factorisation Dynamics and Total Positivity at Several Scales
Series: Analysis, Dynamics, and Applications Seminar
Location: Hybrid, Math 402/Online
Presenter: Jonathan Ramalheira-Tsu, Department of Mathematics, University of Arizona

In this talk, I will present a discrete dynamical system that is closely related to the full Kostant-Toda lattice originally introduced by Ercolani-Flaschka-Singer in 1993. By employing a particular matrix factorisation due to Fomin and Zelevinsky, using so-called Lusztig coordinates and the Loewner-Whitney characterisation of total positivity, I will describe a way of stratifying the overall full dynamics into simpler and more familiar instances of the usual discrete Toda lattice map. The same techniques can be used to provide a simple description of the (continuous-time) full Kostant-Toda lattice in terms of an iterated dynamic evolution on the associated Lusztig coordinates. Going in the other direction, one can perform a spatial discretisation known as ultradiscretisation to obtain a cellular automaton-like evolution which generalises the classical box-ball system of Takahashi and Satsuma (1996) for which the individual strata are comprised precisely of coupled box-ball iterations. If time permits, I will present a beautiful application of this new cellular automaton to the famous Robinson-Schensted-Knuth correspondence which is a fundamental combinatorial bijection with uses in the representation theory of semisimple Lie groups.  This is joint work with Nick Ercolani.

assword: “arizona” (all lower case)